Ell-one and ell-infinity are not reflexive

AC in [A1] supplies the ultrafilter lemma, DC, Hahn--Banach, and Countable [given, A1, L1, L2] Choice needed by [L1] and [L2]. Thus [L1] already refutes reflexivity of ${\mathrm{\ell }}^1$, over both scalar fields.

By [L3], $({\mathrm{\ell }}^1{)}^{\ast }={\mathrm{\ell }}^{\mathrm{\infty }}$. If ${\mathrm{\ell }}^{\mathrm{\infty }}$ were reflexive, [given, L3, L2, step 1.1] the reverse implication in [L2] would make ${\mathrm{\ell }}^1$ reflexive, contradicting step 1.1. Hence ${\mathrm{\ell }}^{\mathrm{\infty }}$ is not reflexive.

Independently, [L4] exhibits the bidual surplus: under the identification [given, L4, A1, step 2.1] $({\mathrm{\ell }}^1{)}^{\ast \ast }=({\mathrm{\ell }}^{\mathrm{\infty }}{)}^{\ast }=ba$, the canonical ${\mathrm{\ell }}^1$ image is only the proper subspace of countably additive charges. This is a concrete failed- surjectivity witness consistent with steps 1.1-2.1.